That depends who are you talking to. It's changing all over so it looks pretty dynamic - one might reserve "static" for a constant rate of change (something like dy/dx = 7.) You might also be talking to people in a context where it's dynamic if the variable things depend on is time, exclusively, and then this would all be static as it describes some static shape, no time anywhere. If you're consulting a variety of source, you might wind up hearing it used both ways?

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

If you have a difficult time with abstract differential equations, you might want to look at classical mechanics courses. Differential equations pop up in physics all the time, basic mechanics' math is not too complicated, and relatively intuitive.

monkey3 wrote:A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives

I suppose that's one way of putting it. I think of them as being equations where the "rates of change" parts are embedded in them rather than isolated on one side or another. It's a bit like parametric equations that way. Differential equations often turn up when rates of change are related to each other as well as to the "thing" that is changing; you can write down that relationship but it's not as simple as "rate of change = function of (x)"

In some cases you can massage a diff. eq. into that form, but in many cases not.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

doogly wrote:That depends who are you talking to. It's changing all over so it looks pretty dynamic - one might reserve "static" for a constant rate of change (something like dy/dx = 7.) You might also be talking to people in a context where it's dynamic if the variable things depend on is time, exclusively, and then this would all be static as it describes some static shape, no time anywhere. If you're consulting a variety of source, you might wind up hearing it used both ways?

Zohar wrote:If you have a difficult time with abstract differential equations, you might want to look at classical mechanics courses. Differential equations pop up in physics all the time, basic mechanics' math is not too complicated, and relatively intuitive.

ucim wrote:

monkey3 wrote:A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives

I suppose that's one way of putting it. I think of them as being equations where the "rates of change" parts are embedded in them rather than isolated on one side or another. It's a bit like parametric equations that way. Differential equations often turn up when rates of change are related to each other as well as to the "thing" that is changing; you can write down that relationship but it's not as simple as "rate of change = function of (x)"

In some cases you can massage a diff. eq. into that form, but in many cases not.

Jose

I am new to this , i don't know much about the applications of differential equations But let me try to understand this

Constant Rate of Change

Rate of Change that is Not Constant

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change")

Then what are differential equations ?

A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives

So a differential equation is any equation which contains the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change")

so far good i guess ?

Last edited by monkey3 on Tue May 09, 2017 7:59 am UTC, edited 1 time in total.

This is why I think it helps to learn this in the context of physics. Let's take, as an example, a spring tied to some sort of mass. That's one of the most common places to start with differential equations in mechanics. Suppose you want to know - how will that mass move around?

You might already know about Newton's third law, that states the force you apply on a (constant) mass is directly proportional to the acceleration that mass will experience: F=ma.

Acceleration is the second derivative of position with relation to time. so you can write a=x`` (I'm writing this notation and assuming we're only going to derive with respect to time). So our equation is now F=mx``.

What's the force that a spring inflicts on a mass? Well, if you have a spring tied to a wall on one end and to a ball at the other, if you pull the ball away from the wall, you'll be working against the spring - the spring will pull you towards the wall. And the more you pull, the harder the spring will pull you towards the wall. Sometimes it's easy to imagine this situations if you imagine how a Loony Toons character would act...

Anyway, so we know the force has to have the opposite sign of the position (if I pull away from the wall, the spring pulls me towards the wall). Now it turns out you can generally say for an "ideal" spring that the formula for the force is F=-kx, where "k" is some sort of constant special for the spring. This is kind of weird reasoning, and we kind of purposefully choose to limit ourselves to something that's relatively easy to solve, that behaves in this way.

So in the end, we get the following equation:-kx=mx``Or put another way:x``=(-k/m)*xSo the question in a differential equation is "What sort of function x(t) would fit here, that would show this relation between acceleration and position?"

There's a bunch of ways to address this. This is a relatively easy case and you can sort of guess the solution, once you have some experience. But that's the very basics of what differential equations are about.

I'm not sure what this means. What is the meaning of "learn" in the context of that you are trying to learn {math field}? You've gone from factoring to differential equations in the course of about a week; I can't imagine learning much more than just "what the words mean" in that time. And there's nothing wrong with that, but it would be helpful to know what your actual goal here is, and also what your prior exposure is (i.e. did you do this stuff in school {mumble} years ago and want to dust it off? Did you just run into some interesting math and became fascinated by the topic?)

btw, for fascinating tidbits, I recommend the numberphile videos on youtube. Cool stuff, very accessible, not much engineering application though (pure math stuff).

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

No, I didn't know that. But in any case, calculus and differential equations are not themselves part of computer science (except in the more esoteric sense). But of course depending on what the program you are writing is supposed to do, you may need the math for that, because you need to understand the subject matter of whatever it is you are programming.

monkey3 wrote:I had to program all these in C programming language , something like extremely difficult . Still struggling ...

I feel for you. Many years ago I set myself the task of learning C from the K&R book. And no computer.

Don't. Ever. Do. That.

Ten years later, after having abandoned that attempt, I tried again, with the Microsoft manual for QuickC, and (later) an actual computer to work on. Went much better.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

And to get to the point of your question, yes, essentially. The point of a differential equation is to find the unknown function, and when we set it up as a differential equation, it's because the things we know about the function are some facts about how its derivatives relate to other variables. This turns out to be super broad and shows up all over.

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

No, I didn't know that. But in any case, calculus and differential equations are not themselves part of computer science (except in the more esoteric sense). But of course depending on what the program you are writing is supposed to do, you may need the math for that, because you need to understand the subject matter of whatever it is you are programming.

monkey3 wrote:I had to program all these in C programming language , something like extremely difficult . Still struggling ...

I feel for you. Many years ago I set myself the task of learning C from the K&R book. And no computer.

Don't. Ever. Do. That.

Ten years later, after having abandoned that attempt, I tried again, with the Microsoft manual for QuickC, and (later) an actual computer to work on. Went much better.

Jose

That is a bit of a useless book to learn C programming language . i had to go through a lot of online tutorials and forums to learn C . Even after that its almost impossible to learn to code from scratch , the best thing would be to study from example codes ... There are many such example codes available on the internet . It would be still kind of cool if you can focus on math programs .

You can easily compile small small math programs in C , make it into an executable and you can even De compile it after that ...I think you can try to crack your own math programs like that ...

morriswalters wrote:I'm curious as to how you ran into something like Numerical Methods without at least a couple of semesters of calculus and differential equations?

I have no idea myself why my computer science syllabus had numerical methods in it in its second semester itself , without any intro to any mathematical subjects except a stupid Set theory in first semester ...

My syllabus sort of looks like this ...

I still have not figured out the relation between the answer of the numerical method and the question ...

doogly wrote:And to get to the point of your question, yes, essentially. The point of a differential equation is to find the unknown function, and when we set it up as a differential equation, it's because the things we know about the function are some facts about how its derivatives relate to other variables. This turns out to be super broad and shows up all over.

Thanks

Last edited by monkey3 on Tue May 09, 2017 8:00 am UTC, edited 1 time in total.

The fact is differential equations very often don't have a clear solution. It's not like, say, finding the prime factors of a number - there are lots of algorithms for that, even though they can take a very long time they'll get you to the right answer.

Most differential equations in real-life applications don't have an easy way to get to a solution, and so using numerical methods you can find a function that's "close enough" to the real solution for what you'll be using it for.

And unfortunately, not every differential equation even has a solution. That said, any equation you are likely to come across almost certainly does, and of course any differential equation describing a real process must have a solution (because, well, something happens). However, many solutions are very complicated and cannot be expressed using simple functions we are familiar with. In these cases especially, we use numerical methods to approximate the solution. A simple example is the Euler method, which can be used to plot points on the solution to certain ordinary differential equations (given an initial value) to arbitrary accuracy.

Given that earlier you had expressed a deficiency in factoring a simple quadratic equation, I am at a loss to understand what good the definition of a differential equation will do you. It would be like a mechanic trying to fix a car without tools. A lot of the work in DE and Calculus is using your knowledge of how to manipulate polynomials to get to forms that can be differentiated or integrated. But good luck any way.

d/dx Is a mathematical operator like '+','-',etc.If you perform this operator on specific function say y=f(x),then the change in 'y' with respect to change in 'x' is dy/dx .also the slope of the graph plotted 'y' vs 'x is dy/dx

ucim wrote:

monkey3 wrote:A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives

I suppose that's one way of putting it. I think of them as being equations where the "rates of change" parts are embedded in them rather than isolated on one side or another. It's a bit like parametric equations that way. Differential equations often turn up when rates of change are related to each other as well as to the "thing" that is changing; you can write down that relationship but it's not as simple as "rate of change = function of (x)"

In some cases you can massage a diff. eq. into that form, but in many cases not.

Jose

I was just wondering what makes this rate of change happen there ? The rate of change dy/dx which is a change in y with respect to change in x ?

Like wind on an object ?

Zohar wrote:The fact is differential equations very often don't have a clear solution. It's not like, say, finding the prime factors of a number - there are lots of algorithms for that, even though they can take a very long time they'll get you to the right answer.

Most differential equations in real-life applications don't have an easy way to get to a solution, and so using numerical methods you can find a function that's "close enough" to the real solution for what you'll be using it for.

You mean its really hard to get that unknown function ?

Eebster the Great wrote:And unfortunately, not every differential equation even has a solution. That said, any equation you are likely to come across almost certainly does, and of course any differential equation describing a real process must have a solution (because, well, something happens). However, many solutions are very complicated and cannot be expressed using simple functions we are familiar with. In these cases especially, we use numerical methods to approximate the solution. A simple example is the Euler method, which can be used to plot points on the solution to certain ordinary differential equations (given an initial value) to arbitrary accuracy.

Like this you plot points ?

morriswalters wrote:Given that earlier you had expressed a deficiency in factoring a simple quadratic equation, I am at a loss to understand what good the definition of a differential equation will do you. It would be like a mechanic trying to fix a car without tools. A lot of the work in DE and Calculus is using your knowledge of how to manipulate polynomials to get to forms that can be differentiated or integrated. But good luck any way.

I don't know if it is such a good idea. You especially can't really deal with differential equations of you don't understand differentiation and integration. I would concentrate on arithmetic, trigonometry, and basic algebra.

Also, can you please stop putting the same pictures over and over again? If you need them for reference have them open in a separate tab, I don't think we need to see them all the time.

Basically? But usually differential equations don't describe something that happens just at one point, but an overall behavior for the function.

The easiest example is probably dy/dx=y. This one you don't need any fancy methods to solve, you just need to know how to differentiate elementary functions and you can figure out the solution yourself.

Have you tried just picking up a book and start working out the problems?

Zohar wrote:Basically? But usually differential equations don't describe something that happens just at one point, but an overall behavior for the function.

The easiest example is probably dy/dx=y. This one you don't need any fancy methods to solve, you just need to know how to differentiate elementary functions and you can figure out the solution yourself.

Have you tried just picking up a book and start working out the problems?

The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change")

But usually differential equations don't describe something that happens just at one point, but an overall behavior for the function.

How is that possible ?

is that y^2 / 2 + c

I don't know any good books to start differential equations . can you suggest some good books ?

No. y is a function that changes with x. What that question is asking is, essentially, what function y(x) is equal to its own derivative dy(x)/dx? Do you know what function that is? Or rather, what family of functions that would be? Importantly, the solution to the differential equation is the function y(x), expressed in terms of x. So obviously the answer cannot be y^2/2 + C, because that's still a function of y; you need a function of x.

Zohar mentioned a similar type of problem:y'' = -ky

This is a second-order differential equation (since there's a second derivative in it). So the question this is asking is: What function y(x) is equal to its second derivative y''(x), times a constant. Can you think of a function that has this property? Try picking a few functions y(x) and sticking them into the equation to see what happens. If you've done some calculus before, you should be able to guess the kind of function that will solve this equation, even without actually using any fancy methods to do so.

Eebster the Great wrote:Also, repeating definitions over and over will not improve your understanding. Instead, you have to practice solving actual problems.

Yes , I am going to start doing that very soon

LaserGuy wrote:

monkey3 wrote:is that y^2 / 2 + c

No. y is a function that changes with x. What that question is asking is, essentially, what function y(x) is equal to its own derivative dy(x)/dx? Do you know what function that is? Or rather, what family of functions that would be? Importantly, the solution to the differential equation is the function y(x), expressed in terms of x. So obviously the answer cannot be y^2/2 + C, because that's still a function of y; you need a function of x.

Zohar mentioned a similar type of problem:y'' = -ky

This is a second-order differential equation (since there's a second derivative in it). So the question this is asking is: What function y(x) is equal to its second derivative y''(x), times a constant. Can you think of a function that has this property? Try picking a few functions y(x) and sticking them into the equation to see what happens. If you've done some calculus before, you should be able to guess the kind of function that will solve this equation, even without actually using any fancy methods to do so.

OK , Thanks

dy/dx=y

I guess i really need to re read that a couple of times to understand

Last edited by monkey3 on Tue May 09, 2017 8:01 am UTC, edited 1 time in total.

Let me give an example. Consider the differential equation dy/dx = 2y. This is a separable differential equation, so you can start out by isolating all the y's on one side and the x's on the other:

dy/y = 2dx

Next, you will want to integrate both sides:

∫dy/y = ∫2dx

If you don't know how to take these integrals, I strongly recommend not worrying about differential equations for quite a while. Math is progressive, which means you need to learn (and master) earlier material before you can tackle later material. You cannot solve differential equations if you don't know how to integrate. You cannot integrate if you don't know algebra. There is no choice but to do this one step at a time. And it is reasonable to expect it to take years to learn, since that's what it takes everybody else.

You can learn some things about differential equations and other areas of math, and you can be interested in them and talk about them, but actually doing it is the real goal, and that is hard.

monkey3 wrote:I guess i really need to re read that a couple of times to understand

monkey3 wrote:OK , i will do everything else other than differential equations .I will keep it aside for some other time .

I think you're doing it wrong.

Whether you are just looking for a refresher (you've done {math field} before but never really got the hang of it and have forgotten it all by now) or it's all new to you, if you are actually trying to learn mathematics (by which I mean are trying to gain the ability to solve mathematical problems, and problems which involve mathematics to solve), then you will not succeed unless you start at the beginning and master each prior step. Yes, there are a few things you can skip, and a few things you can take in a different order, but for the most part, if you can't do algebra, you will not be able to do calculus, even if you understand all the concepts, because you have to do algebra to do calculus. And in order to do algebra, you have to be able to do arithmetic, including fractions and such. If you have trouble adding 2/15 + 7/45, then that trouble will reappear every time you try to do algebra, and it will interfere with your ability to solve algebraic equations, and will slow you down in mastering algebra. With that shaky, you'll have even more trouble with differential calculus, and even more trouble with integral calculus. It won't go away. You won't master the material, and you won't know what it is you are missing. It will just be "hard", and frustrating.

OTOH, if you just want a storybook idea of what the different branches of math are, then sure, you can get that by posting here and reading a few things. But a storybook idea of what calculus is will not let you find the the trajectory of a rocket ship, or anything else that actually uses calculus.

And if you are just trying to learn computer programming, you do not need to know much in the way of math for that. If your syllabus is using lots of math, then either it is geared for people who already know that math (such as engineering students who need to learn programming), or they are doing it wrong.

There is a website (I can't remember it right now but somebody here should recognize it by my description and point you to it) where you can learn programming by actually doing it online... there are many languages you can code in; they give you a partly made program (for a video game) that needs "just this one thing" that you're supposed to code up. That one thing illustrates and uses whatever topic they are teaching, be it a loop, an if statement, whatever. You code up the loop (or whatever) and try it out right there in the web page by playing the game. As you get further along, the projects get more interesting and involved.

Now it's true that certain kinds of programs will need advanced math; writing the code for a rocket engine combustion chamber simulator will require you to understand fluid dynamics, chemistry, thermal transfer, and other stuff. But writing the code for a website or a video game needs none of this.

So, to help you best,

What is your actual overall goal here? Are you trying to learn math for its own sake, or are you just trying to keep up with your programming class? (And is there an actual programming class, or are you studying on your own? Because if so, there may be better options).

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

I am trying to get an overall picture of things in my syllabus , which was from arithmetic to differential equations ...

I have done computer science and i have lost that degree because i lost this computer oriented numerical methods subject .After that i am doing a commerce degree , Right now i have lots of free time to improve all that mathematics .I can now do this for the fun of it .

Computer science required to do mathematics in binary operations , which was hard and uninteresting because of the complexity of the subject back then .

Anyway i decided to do this again from the basics itself , because i have lots of free time .

I was trying to improve most of the things from basics itself , so i made a list of things to improve and it sort of looks like this .

Our text books were dense and stupid , there was no one book that covered all of these .So i decided to look for some online materials .After searching a lot i was able to find some quality stuff online , like websites .

First of all there is an equationThen there is the derivativeThen there is a point slope formula to find the equation of the tangent linePoint slope formula to obtain the tangent line . y=3a2(x-a)+a3Then Plug in the x coordinate into the derivative to get the slope

f'(1) = 3(1)2f'(1) = 3

What this means is that for any value of x=a, the instantaneous slope of f at (a,a3) is 3a2.

Here , i don't really understand some change of terms from f(x) to y , a ... etc

Some differential equations doubt too for one last time before i quit ...I like to keep it like a list of things i should follow before i can focus on individual subjects ...

Is this like a complete list of the types of differential equation ?

Last edited by monkey3 on Tue May 09, 2017 8:01 am UTC, edited 2 times in total.

ucim wrote:And if you are just trying to learn computer programming, you do not need to know much in the way of math for that. If your syllabus is using lots of math, then either it is geared for people who already know that math (such as engineering students who need to learn programming), or they are doing it wrong.

There is a website (I can't remember it right now but somebody here should recognize it by my description and point you to it) where you can learn programming by actually doing it online... there are many languages you can code in; they give you a partly made program (for a video game) that needs "just this one thing" that you're supposed to code up. That one thing illustrates and uses whatever topic they are teaching, be it a loop, an if statement, whatever. You code up the loop (or whatever) and try it out right there in the web page by playing the game. As you get further along, the projects get more interesting and involved.

I don't know if this is the specific one you meant, but I've used Codecademy to fill in some holes in my Python and liked the interface. There are courses in I believe 8 coding languages, beginning with the very basics and advancing to much more complex programming.